Multilabeled Versions of Sperner's and Fan's Lemmas and Applications
Abstract
We propose a general technique related to the polytopal Sperner lemma for proving old and new multilabeled versions of Sperner's lemma. A notable application of this technique yields a cake-cutting theorem where the number of players and the number of pieces can be independently chosen. We also prove multilabeled versions of Fan's lemma, a combinatorial analogue of the Borsuk--Ulam theorem, and exhibit applications to fair division and graph coloring.
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References
1.
M. Alishahi, Colorful subhypergraphs in uniform hypergraphs, Electron. J. Combin., 24 (2017), P1.23.
2.
M. Alishahi, H. Hajiabolhassan, and F. Meunier, Strengthening topological colorful results for graphs, European J. Combin., 64 (2017), pp. 27--44.
3.
N. Alon, Splitting necklaces, Adv. Math., 63 (1987), pp. 247--253.
4.
N. Alon and D. B. West, The Borsuk-Ulam theorem and bisection of necklaces, Proc. Amer. Math. Soc., 98 (1986), pp. 623--628.
5.
M. Asada, F. Frick, V. Pisharody, M. Polevy, D. Stoner, L. H. Tsang, and Z. Wellner, Fair division and generalizations of Sperner- and KKM-type results, SIAM J. Discrete Math., 32 (2018), pp. 591--610, https://doi.org/10.1137/17M1116210.
6.
E. Babson, Meunier Conjecture, preprint, https://arxiv.org/abs/1209.0102, 2012.
7.
P. Bacon, Equivalent formulations of the Borsuk-Ulam theorem, Canad. J. Math., 18 (1966), pp. 492--502.
8.
R. B. Bapat, A constructive proof of a permutation-based generalization of Sperner's lemma, Math. Program., 44 (1989), pp. 113--120.
9.
A. Brown and S. S. Cairns, Strengthening of Sperner's lemma applied to homology theory, Proc. Natl. Acad. Sci. USA, 47 (1961), pp. 113--114.
10.
J. A. De Loera, X. Goaoc, F. Meunier, and N. Mustafa, The discrete yet ubiquitous theorems of Carathéodory, Helly, Sperner, Tucker, and Tverberg, Bull. Amer. Math. Soc., to appear.
11.
J. A. De Loera, E. Peterson, and F. E. Su, A polytopal generalization of Sperner's lemma, J. Combin. Theory Ser. A, 100 (2002), pp. 1--26.
12.
M. De Longueville and R. T. Živaljević, Splitting multidimensional necklaces, Adv. Math., 218 (2008), pp. 926--939.
13.
K. Fan, A generalization of Tucker's combinatorial lemma with topological applications, Ann. of Math. (2), 56 (1952), pp. 431--437.
14.
F. Frick, K. Houston-Edwards, and F. Meunier, Achieving rental harmony with a secretive roommate, Amer. Math. Monthly, 126 (2019), pp. 18--32.
15.
D. Gale, Equilibrium in a discrete exchange economy with money, Internat. J. Game Theory, 13 (1984), pp. 61--64.
16.
C. H. Goldberg and D. B. West, Bisection of circle colorings, SIAM J. Algebraic Discrete Methods, 6 (1985), pp. 93--106, https://doi.org/10.1137/0606010.
17.
C. R. Hobby and J. R. Rice, A moment problem in ${L}_1$ approximation, Proc. Amer. Math. Soc., 16 (1965), pp. 665--670.
18.
S.-N. Lee and M.-H. Shih, A counting lemma and multiple combinatorial Stokes' theorem, European J. Combin., 19 (1998), pp. 969--979.
19.
L. Lovász, Kneser's conjecture, chromatic number and homotopy, J. Combin. Theory Ser. A, 25 (1978), pp. 319--324.
20.
K. L. Nyman and F. E. Su, A Borsuk-Ulam equivalent that directly implies Sperner's lemma, Amer. Math. Monthly, 120 (2013), pp. 346--354.
21.
D. Pálvölgyi, Combinatorial necklace splitting, Electron. J. Combin., 16 (2009), R79.
22.
T. Prescott and F. E. Su, A constructive proof of Ky Fan's generalization of Tucker's lemma, J. Combin. Theory Ser. A, 111 (2005), pp. 257--265.
23.
F. W. Simmons and F. E. Su, Consensus-halving via theorems of Borsuk-Ulam and Tucker, Math. Social Sci., 45 (2003), pp. 15--25.
24.
G. Simonyi, Necklace bisection with one cut less than needed, Electron. J. Combin., 15 (2008), N16.
25.
G. Simonyi, C. Tardif, and A. Zsbán, Colourful theorems and indices of homomorphism complexes, Electron. J. Combin., 20 (2013), P10.
26.
G. Simonyi and G. Tardos, Local chromatic number, Ky Fan's theorem, and circular colorings, Combinatorica, 26 (2006), pp. 587--626.
27.
W. Stromquist, How to cut a cake fairly, Amer. Math. Monthly, 87 (1980), pp. 640--644.
28.
F. E. Su, Rental harmony: Sperner's lemma in fair division, Amer. Math. Monthly, 106 (1999), pp. 930--942.
29.
A. W. Tucker, Some topological properties of disk and sphere, in Proceedings of the First Canadian Mathematical Congress, Montreal, Canada, 1945, pp. 285--309.
30.
D. R. Woodall, Dividing a cake fairly, J. Math. Anal. Appl., 78 (1980), pp. 233--247.
31.
R. T. Živaljević, Oriented matroids and Ky Fan's theorem, Combinatorica, 30 (2010), pp. 471--484.
Information & Authors
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Published In
SIAM Journal on Applied Algebra and Geometry
Pages: 391 - 411
DOI: 10.1137/18M1192548
ISSN (online): 2470-6566
Copyright
© 2019, Society for Industrial and Applied Mathematics.
History
Submitted: 6 June 2018
Accepted: 23 April 2019
Published online: 9 July 2019
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National Science Foundation https://doi.org/10.13039/100000001 : DMS-1440140
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